PhaseLift

PhaseLift is a convex optimization method for solving the phase retrieval problem, which aims to recover a complex-valued signal x ∈ C^n from magnitude-only measurements y_k = |⟨a_k, x⟩|^2 for k = 1,...,m. The approach recasts the problem by lifting it to a higher-dimensional space: define the rank-one matrix X = x x*, where x* denotes the conjugate transpose. The measurements become linear in X: y_k = a_k* X a_k, where a_k are known measurement vectors.

PhaseLift solves a semidefinite program to recover X and hence x. The standard formulation minimizes the trace of X subject to X being positive semidefinite and satisfying the measurement constraints a_k* X a_k = y_k for all k. The trace objective promotes low rank, and the convex feasibility around the measurements allows exact recovery under suitable conditions. In the presence of noise, the equalities are replaced by fitting constraints or a robust loss, yielding a stable reconstruction.

The method typically requires a number of measurements m that scales on the order of n log n for certain random measurement models, and it provides exact recovery with high probability in ideal settings. PhaseLift is computationally intensive for large n since it operates on n-by-n positive semidefinite matrices, but several scalable variants and solver techniques have been developed, including low-rank factorization approaches and first-order methods.

Variants and related work include PhaseMax, PhaseCut, and nonconvex approaches such as Wirtinger flow. PhaseLift has been applied and studied in optical imaging, X-ray crystallography, and quantum state tomography, where recovering a signal from magnitude information is essential.